Chapter 11

Describe the given set with a single equation or with a pair of equations. The circle of radius 2 centered at (0,0,0) and lying in the a. \(x y\) -plane b. \(y z\) -plane c. \(x z\) -plane

Find the areas of the parallelograms whose vertices are given. $$A(0,0,0), \quad B(3,2,4), \quad C(5,1,4), \quad D(2,-1,0)$$

Sketch the surfaces ASSORTED $$x^{2}+z^{2}=1$$

If \(\overrightarrow{A B}=\mathbf{i}+4 \mathbf{j}-2 \mathbf{k}\) and \(B\) is the point \((5,1,3),\) find \(A\)

find the distance from the point to the plane. $$(2,-3,4), \quad x+2 y+2 z=13$$

Describe the given set with a single equation or with a pair of equations. The circle of radius 2 centered at (0,2,0) and lying in the a. \(x y\) -plane b. \(y z\) -plane c. plane \(y=2\)

Find the areas of the parallelograms whose vertices are given. $$A(1,0,-1), \quad B(1,7,2), \quad C(2,4,-1), \quad D(0,3,2)$$

Sketch the surfaces ASSORTED $$16 y^{2}+9 z^{2}=4 x^{2}$$

If \(\overrightarrow{A B}=-7 \mathbf{i}+3 \mathbf{j}+8 \mathbf{k}\) and \(A\) is the point \((-2,-3,6),\) find \(B .\)

find the distance from the point to the plane. $$(0,0,0), \quad 3 x+2 y+6 z=6$$

Describe the given set with a single equation or with a pair of equations. The circle of radius 1 centered at (-3,4,1) and lying in a plane parallel to the a. \(x y\) -plane b. \(y z\) -plane c. \(x z\) -plane

Find the areas of the triangles whose vertices are given. $$A(0,0), \quad B(-2,3), \quad C(3,1)$$

Sketch the surfaces ASSORTED $$z=-\left(x^{2}+y^{2}\right)$$

Linear combination \(\quad\) Let \(\mathbf{u}=2 \mathbf{i}+\mathbf{j}, \mathbf{v}=\mathbf{i}+\mathbf{j},\) and \(\mathbf{w}= \mathbf{i} - \mathbf{j}\). Find scalars \(a\) and \(b\) such that \(\mathbf{u}=a \mathbf{v}+b \mathbf{w} .\)

find the distance from the point to the plane. $$(0,1,1), \quad 4 y+3 z=-12$$

Describe the given set with a single equation or with a pair of equations. The line through the point (1,3,-1) parallel to the a. \(x\) -axis b. \(y\) -axis c. \(z\) -axis

Find the areas of the triangles whose vertices are given. $$A(-1,-1), \quad B(3,3), \quad C(2,1)$$

Sketch the surfaces ASSORTED $$y^{2}-x^{2}-z^{2}=1$$

find the distance from the point to the plane. $$(2,2,3), \quad 2 x+y+2 z=4$$

Describe the given set with a single equation or with a pair of equations. The set of points in space equidistant from the origin and the point (0,2,0)

Find the areas of the triangles whose vertices are given. $$A(-5,3), \quad B(1,-2), \quad C(6,-2)$$

Sketch the surfaces ASSORTED $$4 y^{2}+z^{2}-4 x^{2}=4$$

Linear combination \(\quad\) Let \(\mathbf{u}=\langle 1,2,1\rangle, \mathbf{v}=\langle 1,-1,-1\rangle\) \(\mathbf{w}=\langle 1,1,-1\rangle,\) and \(\mathbf{z}=\langle 2,-3,-4\rangle .\) Find scalars \(a, b,\) and \(c\) such that \(\mathbf{z}=a \mathbf{u}+b \mathbf{v}+c \mathbf{w}\).

Find the work done by a force \(\mathbf{F}=5 \mathbf{i}\) (magnitude \(5 \mathrm{N}\) ) in moving an object along the line from the origin to the point (1,1) (distance in meters).

find the distance from the point to the plane. $$(0,-1,0), \quad 2 x+y+2 z=4$$

Describe the given set with a single equation or with a pair of equations. The circle in which the plane through the point (1,1,3) perpendicular to the \(z\) -axis meets the sphere of radius 5 centered at the origin

Find the areas of the triangles whose vertices are given. $$A(-6,0), \quad B(10,-5), \quad C(-2,4)$$

Sketch the surfaces ASSORTED $$x^{2}+y^{2}=z$$

Linear combination \(\quad\) Let \(\mathbf{u}=\langle 1,2,2\rangle, \mathbf{v}=\langle 1,-1,-1\rangle\) \(\mathbf{w}=\langle 1,3,-1\rangle,\) and \(\mathbf{z}=\langle 2,11,8\rangle .\) Write \(\mathbf{z}=\mathbf{u}_{1}+\mathbf{u}_{2}+\mathbf{u}_{3}\) where \(\mathbf{u}_{1}\) is parallel to \(\mathbf{u}, \mathbf{u}_{2}\) is parallel to \(\mathbf{v},\) and \(\mathbf{u}_{3}\) is parallel to w. What are \(\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3} ?\)

Pacific's Big Boy locomotive could pull 6000 -ton trains with a tractive effort (pull) of \(602,148 \mathrm{N}\) \((135,375 \mathrm{lb})\). At this level of effort, about how much work did Big Boy do on the (approximately straight) \(605-\mathrm{km}\) journey from San Francisco to Los Angeles?

find the distance from the point to the plane. $$(1,0,-1), \quad-4 x+y+z=4$$

Describe the given set with a single equation or with a pair of equations. The set of points in space that lie 2 units from the point (0,0,1) and, at the same time, 2 units from the point (0,0,-1)

Find the areas of the triangles whose vertices are given. $$A(1,0,0), \quad B(0,2,0), \quad C(0,0,-1)$$

a. Express the area \(A\) of the cross-section cut from the ellipsoid $$x^{2}+\frac{y^{2}}{4}+\frac{z^{2}}{9}=1$$ by the plane \(z=c\) as a function of \(c .\) (The area of an ellipse with semiaxes \(a\) and \(b\) is \(\pi a b .\) ) b. Use slices perpendicular to the \(z\) -axis to find the volume of the ellipsoid in part (a). c. Now find the volume of the ellipsoid $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 $$ Does your formula give the volume of a sphere of radius \(a\) if \(a=b=c ?\)

When solving you may need to use a calculator or a computer. Velocity \(\quad\) An airplane is flying in the direction \(25^{\circ}\) west of north at \(800 \mathrm{km} / \mathrm{h} .\) Find the component form of the velocity of the airplane, assuming that the positive \(x\) -axis represents due east and the positive \(y\) -axis represents due north.

Find the distance from the plane \(x+2 y+6 z=1\) to the plane \(x+2 y+6 z=10\)

Write inequalities to describe the sets The slab bounded by the planes \(z=0\) and \(z=1\) (planes included)

Find the areas of the triangles whose vertices are given. $$A(0,0,0), \quad B(-1,1,-1), \quad C(3,0,3)$$

Find the distance from the line \(x=2+t, y=1+t\) \(z=-(1 / 2)-(1 / 2) t\) to the plane \(x+2 y+6 z=10\)

Write inequalities to describe the sets. The solid cube in the first octant bounded by the coordinate planes and the planes \(x=2, y=2,\) and \(z=2\)

Find the areas of the triangles whose vertices are given. $$A(1,-1,1), \quad B(0,1,1), \quad C(1,0,-1)$$

Show that the volume of the segment cut from the paraboloid $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=\frac{z}{c}$$ by the plane \(z=h\) equals half the segment's base times its altitude.

Find the angles between the planes. $$x+y=1, \quad 2 x+y-2 z=2$$

Write inequalities to describe the sets. The half-space consisting of the points on and below the \(x y\) -plane

Find the volume of a parallelepiped with one of its eight vertices at \(A(0,0,0)\) and three adjacent vertices at \(B(1,2,0), C(0,-3,2)\) and \(D(3,-4,5)\).

a. Find the volume of the solid bounded by the hyperboloid $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1$$ and the planes \(z=0\) and \(z=h, h>0\) b. Express your answer in part (a) in terms of \(h\) and the areas \(A_{0}\) and \(A_{h}\) of the regions cut by the hyperboloid from the planes \(z=0\) and \(z=h\) c. Show that the volume in part (a) is also given by the formula $$ V=\frac{h}{6}\left(A_{0}+4 A_{m}+A_{h}\right) $$ where \(A_{m}\) is the area of the region cut by the hyperboloid from the plane \(z=h / 2\)

Find the angles between the planes. $$5 x+y-z=10, \quad x-2 y+3 z=-1$$

Write inequalities to describe the sets. The upper hemisphere of the sphere of radius 1 centered at the origin

Triangle area Find a \(2 \times 2\) determinant formula for the area of the triangle in the \(x y\) -plane with vertices at \((0,0),\left(a_{1}, a_{2}\right),\) and \(\left(b_{1}, b_{2}\right) .\) Explain your work.

Plot the surfaces in Exercises over the indicated domains. If you can, rotate the surface into different viewing positions. $$z=y^{2}, \quad-2 \leq x \leq 2, \quad-0.5 \leq y \leq 2$$